and fine information about the linearized equation is available.
Similar statements can be drawn for generic choices of rapidly decaying, radially decreasing positive building blocks \(U(|y|)\) .
Vortices (REF ) have been used to construct solutions with concentrated vorticities in \(\mathbb {R}^2\)  in [1]}
and helicoidal travelling waves in [2]}. Linearization-stability in \(2d\) -Euler and with a third dimension added
as a vortex filament is a delicate matter, see for instance
[3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
